Circular Measure - Form 5 Mathematics

Introduction

• Circular Measure is the first chapter in Form 5 Mathematics.
• Key areas covered include conversion between radian and degree, finding arc length, area of sector, and area of segment.

Key Concepts

1. Conversion between Radian and Degree

• Radian and Degree are two units for measuring angles.
• Conversion:
  • 360 degrees = 2π radians
  • 180 degrees = π radians
  • 90 degrees = π/2 radians
  • 45 degrees = π/4 radians
• Formula:
  • Degree to Radian: degree × π/180
  • Radian to Degree: radian × 180/π

2. Finding Arc Length

• Arc Length (S): S = rθ
  • r = radius
  • θ = angle in radians

3. Area of a Sector

• Formula: Area = 1/2 r²θ
  • r = radius
  • θ = angle in radians

4. Area of a Segment

• Formula: Area = 1/2 r² (θ - sin(θ))
  • θ in radians for the first part
  • θ in degrees for trigonometric functions (sine, cosine, tangent)

Examples and Applications

1. Conversion Examples

• Example: Convert π/8 radians to degrees:
  • π/8 × 180/π = 22.5 degrees
• Example: Convert 1.04 radians to degrees:
  • 1.04 × 180/π = 59.58 degrees
• Example: Convert 150.5 degrees to radians:
  • 150.5 × π/180 = 2.626 radians
• Example: Convert 220 degrees to radians:
  • 220 × π/180 = 3.8402 radians

2. Finding Arc Length

• Example: For a sector with radius 7 cm and arc length 14 cm, find the angle in radians and degrees.
  • S = rθ → 14 = 7θ → θ = 2 radians
  • Convert to degrees: 2 × 180/π ≈ 114.58 degrees

3. Application in Real-World Problems

• Example: Calculate the length of a bicycle chain given the circumferences of front and back cranks (50.8 cm and 30.5 cm respectively) and angles.
• Example: Calculate the area of a water pipe’s cross-section, given internal radius and horizontal width.

Practice Problems

Simple Conversion

• Convert each angle given in degrees to radians and vice versa.

Arc Length Calculation

• Find the arc length given the radius and angle in radians.

Area Calculations

• Calculate the area of a sector and segment using the given formulas.

Perimeter and Area of Composite Shapes

• Find the perimeter and area of complex shapes involving circular segments and sectors.

Important Notes

• Always ensure the angle is in radians when using formulas for arc length and area of a sector/segment.
• Use degrees for trigonometric functions (sine, cosine, tangent) to avoid incorrect answers.

Conclusion

• Practice the given examples to master the conversion and calculation techniques in circular measure.
• Verify your results by checking units and consistency in using radians and degrees.